Andréa V. Rocha

Improved estimators for a general class of beta regression models

In this article, we extend the beta regression model proposed by Ferrari and Cribari-Neto (2004), which is generally useful in situations where the response is restricted to the standard unit interval in two different ways: we let the regression structure to be nonlinear, and we allow a regression structure for the precision parameter (which may also be nonlinear). We derive general formulae for second order biases of the maximum likelihood estimators and use them to define bias-corrected estimators. Our formulae generalize the results obtained by Ospina et al. (2006), and are easily implemented by means of supplementary weighted linear regressions. We compare, by simulation, these bias-corrected estimators with three different estimators which are also bias-free to second order: one analytical, and two based on bootstrap methods. The simulation also suggests that one should prefer to estimate a nonlinear model, which is linearizable, directly in its nonlinear form. Our results additionally indicate that, whenever possible, dispersion covariates should be considered during the selection of the model, as we exemplify with two empirical applications. Finally, we also present simulation results on confidence intervals.

Skewness of maximum likelihood estimators in dispersion models

We introduce the dispersion models with a regression structure to extend the generalized linear models, the exponential family nonlinear models (Cordeiro and Paula, 1989) and the proper dispersion models (Jorgensen, 1997a). We provide a matrix expression for the skewness of the maximum likelihood estimators of the regression parameters in dispersion models. The formula is suitable for computer implementation and can be applied for several important submodels discussed in the literature. Expressions for the skewness of the maximum likelihood estimators of the precision and dispersion parameters are also derived. In particular, our results extend previous formulas obtained by Cordeiro and Cordeiro (2001) and Cavalcanti et al. (2009). A simulation study is performed to show the practice importance of our results.

Asymptotic adjustments of Pearson residuals in exponential family nonlinear models

ABSTRACT In this work we define a set of corrected Pearson residuals for continuous exponential family nonlinear models that have the same distribution as the true Pearson residuals up to order , where n is the sample size. Furthermore, we also introduce a new modification of the Pearson residuals, which we call PCA Pearson residuals, that are approximately uncorrelated. These PCA residuals are new even for the generalized linear models. The numerical results show that the PCA residuals are approximately normally distributed, thus improving previous results by Simas and Cordeiro [Adjusted Pearson residuals in exponential family nonlinear models. J Stat Comput Simul. 2009;79:411–425]. These numerical results also show that the corrected Pearson residuals approximately follow the same distribution as the true residuals, which is a considerable improvement with respect to the Pearson residuals and also extends the previous work by Cordeiro and Simas [The distribution of Pearson residuals in generalized linear models. Comput Stat Data Anal. 2009;53:3397–3411].